Translation:He takes his wife along on a business trip.
Well "nemen" is specifically "to take". If it was "to bring", it should be "brengen". So it depends how literally or loosely you want to translate. Personally, I think changing it to "bring", when the original said "take" is taking some liberties with the text. In English, it's acceptable to say either - and I suspect in Dutch too. But when translating, it's good practice to stick as closely as possible to the original, unless it results in an obviously clumsy or unnatural phrase in the target language. Also, even in English, I think whether it's "bring" or "take" depends on the location of the speaker. If they are already at the destination of the business trip, or going too, they would say someone is bringing his wife. If they are stopping at home, they would say he's taking her.
You can add 'mee' at the end, but it's not necessary, this word order is also right. 'Hij neemt zijn vrouw op zakenreis mee' is also good, but I feel 'Hij neemt zijn vrouw mee op zakenreis' is more natural. Maybe it has something to do with the 'thing' that you're taking with you? 'zijn vrouw' is the object here that is done something with, so the verb "wraps" around it.
No, the "him" is not redundant or optional in correct English. The only way you could avoid it would be to substitute "along" for "with". If you say: "He takes his wife along on business trips", the "with him" part is implicit (although it would not be incorrect still to include it).